Finding partitions $\{A,B\}$ of the set $\mathbb N_n:=\{1,...,n\}$ such that $\Big|\prod_{i \in A}p_i-\prod_{i\in B}p_i\Big|=1$

Question

For a fixed $n$ , for what partitions $\{A,B\}$ of the set $\mathbb N_n:=\{1,...,n\}$ do we have

$\Big|\prod_{i \in A}p_i-\prod_{i\in B}p_i\Big|=1$ ? , where $p_m$ denotes the $m$th prime for example $p_1=2,p_2=3...$ etc.

When the quantity is not $1$ , I can conclude that $p_{n+1} \le \Big|\prod_{i \in A}p_i-\prod_{i\in B}p_i\Big| $ this is why I am

interested in it . Also I would like to ask , for every integer $n>1$ does there exist a partition $\{A,B\}$

of $\mathbb N_n:=\{1,...,n\}$ such that $\Big|\prod_{i \in A}p_i-\prod_{i\in B}p_i\Big|=1$ ? If not then for which $n$ there does not exist such a partition ?

Answer 1

There is some information about this problem in Guy, Lacampagne, and Selfridge, Primes at a glance, Mathematics of Computation 48, #177 (January 1987) 183-202, available here. See in particular Table 4, also the reference to D H Lehmer, On a problem of Stormer, Illinois J Math 8 (1964) 59-79, which should be available here.

EDIT: Carl Pomerance, Ruth-Aaron numbers revisited, in the book, Paul Erdos and his Mathematics, 567-579, writes that it is conjectured that 714, 715 is the last pair of consecutive numbers whose product is the product of the first $k$ primes for some $k$. The paper is here.